Official Solution:


Notice that we can get the equation of line \(k\) which is perpendicular to line \(y=2x\) if we know ANY point that line \(k\) passes through. So, to get the equation of line \(k\) we need the values of \(a\) and \(b\).

(1) \(a = -b\). Not sufficient.

(2) \(a - b = 1\). Not sufficient.

(1)+(2) \(a = -b\) and \(a - b = 1\), we have two distinct linear equations with two unknowns so we can solve for \(a\) and \(b\). Sufficient.


Answer: C
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hi Bunuel

can't A suffice to find solution? y=2x is given as going from quad 3 through origin to quad 1 with slope 2.

slope of the perped line is -1/2, relatioship between points is given as whenever a is positive the b bears the same value but with opposite sign (10,-10 or -10;10) - hence the second line has to go through the origin from quad 2 to quad 4?

thanks

Why should line k go through the origin?
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Hi friends,

the line y=2x surely passes through the origin as the coordinates (0,0) satisfies the equation.

Now coming to the point, the solution says both the statements are required, but if you take the slope of second line as -1/2 then the equation for is

-1/2=(y-b)/(x-a)
or, 2y= -x +(a+b)
or, y=(-1/2)x+(a+b)/2

Now, applying statement 1 (a=-b)to this equation;

y=(-1/2)x+{a+(-a)}/2
or, y=(-1/2)x+ 0/2
or, y=(-1/2)x

So, statement 1 seems sufficient (contradiction with the actual answer)

Check the diagram below:



As you can see if a = 1 and b = -1 we have different line perpendicular to y = 2x than if a = 2 and b = -2.



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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Bunuel,

If the intention is merely to find values of a and b then what is the significance of "Perpendicularity" here? Please explain

The question asks to find the equation of k that is perpendicular to line y=2x and passes through point (a,b). We need to find the values of a and b to answer the question.
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I think that this is a high-quality question and the explanation is very clear. Thanks for posting.

S1) take different values for a,b that meet the given statement (a=-b) and form lines equations of lines with slope -1/2. You can see that for different values of (a,b), you get different equations.

S2) same as statement 1

Both) Both equations give a point a=1/2 and b=-1/2. Which hivea only one equation 2x+4y=-1

So the ans is C. Hope fully my logic ia correct.

Posted from my mobile device

i want to apologize, as i am very very weak at coordinate geometry.

First, please give a suggestion how can i learn coordinate geometry.

2nd, isn't it obvious in option A that b=5, a= -5; b= 1 a= -1; b=0, a=0; b=1, a= -1?

sorry again if i ask something foolish
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For coordinate geometry check this: math-coordinate-geometry-87652.html

For more on this question check here: what-is-the-equation-of-the-line-k-that-is-perpendicular-to-line-y-69481.html
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Hi Bunuel,

A little clarification
equation of line perpendicular to y-2x=0 will be -2y-x+c=0 ??
In order to find equation of a line perpendicular to ax+by+c=0.....interchange coefficients of x and y and change the sign in between....so equation of line perpendicular to ax+by+c=0 will be bx-ay+k=0.
Is this approach correct?

I think this is a high-quality question and I agree with explanation.

Hi Bunuel,

If the question were to be a slightly different one, would the below Statement be sufficient?

Question stem: Find equation of the line K, perpendicular to a line y= -1/2x and passes through point (a,b).

statement 1) 2a=b

I understood the point that given a slope + either the y-intercept or a specific point, we can plot the specific line. Where I am getting confused is whether we can still identify the line by having the slope and a generic equation such as 2a=b.

When I tried plotting several lines with the slope = 2 and with different y-intercepts, I could not come up with any point that satisfy 2a=b and any other lines except y=2x+0.

Would you please help me clear this point? Thanks!

I do not understand why we cannot say that stat I implies a=b=0, hence being sufficient while statement II implies basically all consecutive integers, not being sufficient. Can someone please explain?

a = -b does not necessarily means that a = b= 0. Why not a= 1 and b = -1?
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Hi, Can anyone tell me what will be the final equation of the line once we know what a and b are?

From a=-b and a-b=1, we can get that a = 1/2 and b = -1/2. So, we know that k passes through the point (1/2, -1/2) and is perpendicular to line y = 2x (slope = 2)

The two lines are perpendicular if and only if the product of their slopes is -1, so m*2 = -1 and m = -1/2 (the slope of line k).

Finally, the equation of a straight line that passes through a point \(P_1(x_1, y_1)\) with a slope m is: \(y-y_1=m(x-x_1)\). Substitute: \(y-(-\frac{1}{2})=-\frac{1}{2}(x-\frac{1}{2})\) --> \(y = -\frac{x}{2} - \frac{1}{4}\).

Check below:




For Coordinate Geometry check:

24. Coordinate Geometry

• Theory
  Math: Coordinate Geometry
  Beauty of Coordinate geometry
  Dealing with Tangents on the GMAT
  
  • Questions
    DS Questions
    PS Questions

For more check:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.



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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. i read the whole thread, and still didn't understand the explanation. I do understand that we don't need to find the value of a and b, question is why would we discard 1） and 2) as insufficient, what's the reasoning behind it.